A Simplified Approach to Estimating Parameter of the GARCH (1,1) Model

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Published: Aug 22, 2019
DOI: https://doi.org/10.14416/j.asep.2019.06.001
Keywords:
Quasi-newton method Initial values Contour plots GARCH (1,1) model Log-likelihood function Volatility

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Farida Farida
Department of Management, Faculty of Economics, Darma Persada University, Duren Sawit, East Jakarta, Indonesia
Nifatamah Makaje
Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus, Thailand
Aniruth Phon-On
Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus, Thailand

Abstract

The paper aims to present a method of parameter estimation of the GARCH (1,1) model. This estimation problem involves computing the parameter estimates by maximizing the log-likelihood function. The Quasi-Newton method and an appropriate starting point for the iterations were used. The idea of contour plots was applied to find the initial values of the parameters for the iterative process. The method presented in this paper was illustrated using the Paris Bourse stock exchange and Thailand rubber data. The estimated parameters were compared to the values obtain using the Excel’s Solver, in order to know accurateness of the method presented. The results showed that, apart from the Excel’s Solver, applying the Quasi-Newton method together with the idea of contour plots provides an appropriate approach for parameter estimation of the GARCH (1,1) model. The parameters obtained was then used to estimate the volatility of the Paris Bourse stock exchange data and the results obtained was consistent with a previous study.

Article Details

How to Cite
Farida, F., Makaje, N., & Phon-On, A. (2019). A Simplified Approach to Estimating Parameter of the GARCH (1,1) Model. Applied Science and Engineering Progress, 12(3), 158–163. https://doi.org/10.14416/j.asep.2019.06.001
Issue
Vol. 12 No. 3 (2019)
Section
Research Articles

References

[1] S. Saejiang, W. Pornwiriyamongkol, and D. McNeil, “Time series analysis of banking share returns in Thailand,” Songklanakarin Journal of Science and Technology, vol. 23, no. 3, pp. 444–448, Mar. 2001.

[2] L. H. Ederington and W. Guan, “Forecasting volatility,” Journal of Futures Markets, vol. 25, no. 5, pp. 465–490, May 2005.

[3] A. E. M. Ahmed and S. Z. Suliman, “Modelling stock market volatility using GARCH models evidence from Sudan,” International Journal of Business and Social Science, vol. 2, no. 3, pp.114–128, Mar. 2011.

[4] V. Y. Naimy, “Parameterization of GARCH (1,1) for Paris stock market,” American Journal of Mathematics and Statistics, vol. 3, no. 6, pp. 357–361, Jun. 2013.

[5] P. Posedel, “Properties and estimation of GARCH (1,1) model,” Metodoloski Zvezki, vol. 2, no. 2, pp. 243–257, Feb. 2013.

[6] T. G. Andersen, T. Bollerslev, P. F. Christoffersen, and F. X. Diebold, Volatility and Correlation Forecasting. Amsterdam, Netherlands: North-Holland Press, 2006.

[7] Y. Yang. (2012, Dec.). Parameter estimation of GARCH model. The State University of New York. New York [Online]. Available: http://www.ams.stonybrook.edu/~yiyang/research/computational_finance/Parameter_Estimation_of_GARCH_Model.pdf

[8] F. Qi and S. Chen, “Complete monotonicity of logarithmic mean,” Mathematical Inequalities and Applications, vol. 10, no. 4, pp. 799–804, Oct. 2007.

[9] H. Lee, S. Chen, and H. Kang, “A study of generalized reduced gradient method with different search directions,” Measurement Management Journal, vol. 1, no. 1, pp. 25–38, 2004.

[10] H. K. Lo and W. Y. Szeto, “Planning transport network improvments over time,” in Urban and Regional Transportation Modeling : Essays in Honor of David Boyce, UK: Edward Elgar Publishing, Inc, 2003, pp. 157–176.

[11] J. Arneric and A. Rozga, “Numerical optimization within vector of parameters estimation in volatility models,” International Journal of Information and Communication Engineering, vol. 3, no. 1, pp. 112–116, 2009.

[12] R. L. Burden and J. D. Faires, Numerical Analysis, 9th ed., USA: Brooks/Cole, 2011, pp. 144–146.

[13] G. Storvik. (2011, Feb.). Numerical Optimization of likelihoods: Additional Literature for STK2120. University of Oslo [Online]. Available: https://folk.ntnu.no/bo/ST2202/2008h/numerisk.opt.storvik.pdf

[14] M. L. Rizzo, Statistical Computing with R. USA: Chapman & Hall, 2008, pp. 106.

[15] C. Alexander, Practical Financial Econometrics. UK: John Wiley & Sons, 2008, pp. 131–198.

[16] J. C. Hull, Options, Futures, and Other Derivatives. 7th ed., New Jersey: Pearson Prentice Hall, 2009, pp. 469–486.

[17] H. Baudchon. (2017, Jun.). Overview of the France Economy. BNP PARIBAS, Bangkok [Online]. Available: http://economic-research.bnpparibas.com/Views/DisplayPublication.aspx?type=document&IdPdf=29951

[18] European Economy Team. (2009). Economic Crisis in Europe: Causes, Consequences and Responses. European Communities, Italy [Online]. Available : http://ec.europa.eu/economy_finance/publications/pages/publication15887_en.pdf

[19] E. D. Vrijer and B. Yontcheva. (2009, Jul.). IMF Survey : France : Less Severe Recession but Tepid Recovery. International Monetary Fund, Washington, DC [Online]. Available: https://www.imf.org/en/News/Articles/2015/09/28/04/53/socar073109a

[20] Wolfram Company. (2017, Jul.). Density and Contour Plots. Wolfram Company, Illinois, USA [Online]. Available: http://reference.wolfram.com/language/tutorial/DensityAndContourPlots.html